The pink colour comes from blood close to the surface showing through skin that has been repeatedly abraded and healed. Scar tissue is thinner and less pigmented than intact skin, so an animal accumulates pink as it accumulates damage. Body temperature and excitement can deepen the flush temporarily, in the same way that thin skin flushes in other mammals.
Long-term work in the Brazilian Amazon found that adult males carry far more tooth-rake scarring than females and are correspondingly pinker, and that the most heavily marked males are also the largest. That points to conspicuous colour functioning as an honest signal of fighting history rather than as camouflage.
Caution is warranted about how strongly this is established. The correlation between scarring, size and colour is documented, but the reproductive payoff of being pink has not been measured directly, and juveniles and some adults remain grey throughout.